1. Field of the Invention
The present invention relates to an optical measuring apparatus, and more specifically it relates to a wavefront measurement apparatus.
2. Description of the Related Art
Wavefront measurements are used to test the quality of optical surfaces and evaluate optical performance of optical elements. Wavefront measurements are also used for alignment of high-accuracy optical systems. A wavefront is the locus (a line, or in a wave propagating in 3 dimensions, a surface) of points on which all light rays have the same phase. The simplest form of a wavefront is that of a plane wave, where rays of light are parallel to each other and strike a sensor with a planar wavefront. Propagation of a wavefront through optical elements, such as lenses and mirrors, generally changes the shape of the wavefront due to lens thickness, imperfections in surface morphology, variations in refractive index, and other factors. These changes in the shape of the wavefront are known as aberrations. Thus, knowledge of the wavefront profile and correction of aberrations thereof are very important when designing optical elements, and evaluating the performance of newly designed optical systems. For example, before assembling a complete imaging system, it is necessary to verify performance of each optical (unit lens) included in such a system. Since each unit lens or single lens itself may have certain aberration, it is necessary to control the image quality of imaging lenses at high precision.
A conventional method of measuring the wavefront quality of a light beam employs interferometric wavefront sensors in which spatial filtering of a small portion of the light source beam is used to produce a speherical reference wave that is subsequently combined with the original wavefront to produce an interferogram. As it is will understood in the art, interference fringes in the inteferogram can be analyzed to evaluate the quality of the light beam. However, dividing the original beam and recombining it with the reference beam tends to introduce system aberrations, such as optical-path errors and improper alignment. Another conventional method of wavefront measurement uses Shack-Hartmann wavefront sensors, which do not require dividing and recombining the original beam.
Shack-Hartmann wavefront sensors (SHWFS) are commonly used as a large dynamic range wavefront sensor. One of the most basic and generally used configurations for a SHWFS sensor consists of a lenslet array and an optical detector (typically a CCD camera). The Shack-Hartmann wavefront sensor divides the wavefront of an incident beam being measured into a plurality of beamlets by using a two-dimensional lenslet array. Each lenslet in the lenslet array generates a separate and independent focus (spot) on the surface of the optical detector. The centroid position of each spot is displaced by wavefront aberrations between a reference and distorted beam. Therefore, wavefront measurement by the SHWFS sensor is based on the measurements of the local slopes of an aberrated wavefront relative to a reference (plane) wavefront. Generally, the wavefront estimation procedure may be categorized as either zonal or modal, depending on whether the phase is presented like a number of local slopes of the wavefronts or in terms of coefficients of some modal functions determined on the whole aperture. In the latter, displacements of focal sport can be represented in terms of Zernike polynomials. There are several advantages to using SHWFS over interferometric counterparts. SHWFS have greater dynamic range than interferometric sensors. The incident radiation does not have to be coherent. Since the SHWFS can acquire all of the wavefront information from a single image, exposure times can be short, which reduces sensitivity to vibration. More importantly, both irradiance and phase distributions can be obtained with an SHWFS.
FIG. 1 illustrates an example of the configuration of a wavefront measurement system using SHWFS. In FIG. 1, a laser 1000, a ND Filter 1001, a beam expander 1002, a lens 1004, test optics (i.e., a sample) 1005, a lens 1006, a lens 1007, a lenslet array (micro lens array) 1008, a CCD sensor 1010, a data analyzer 1009 are arranged in a predetermined manner to characterize the effects that the sample exerts on a wavefront of light traveling therethrough.
The optical configuration of the SHWFS is illustrated with more detail in FIG. 2. In FIG. 2, locations of focal spots (2500 to 2503) on the detector array 2010 are dependent on a local tilt of the incoming wavefront. The local tilt of the wavefront is caused by aberrations due to the test optics 1005 in FIG. 1. The local tilt is calculated by variation of focal spot location. The wavefront can be reconstructed by using the local tilt information obtained from all lenslets of the lenslet array 1008.
When the amount of wavefront deviation is less than a dynamic range of SHWFS, positions of each spot on the detector array 2010 can be detected separately. If the wavefront deviation exceeds the dynamic range of SHWFS, and the focal spots on the detector array 2010 cross each other, then SHWFS cannot analyze the wavefront anymore. FIG. 3 illustrates the situation in which the wavefront deviation exceeds the dynamic range of SHWFS. In FIG. 3, focal spot 3500 is located outside of the detector array 2010, and focal spots 3501, 3502 and 3503 are located on the surface of the detector array 2010. However, focal spots 3502 and 3503 are crossed (i.e., the beams forming these spots overlap each other and are focused at crossed locations). This situation is caused by the large aberrated wavefront incident on the lenslet array 1008.
FIG. 4A shows output data from the detector array 2010 with large aberrated wavefront. Focal spots at the outer region (denoted by a square 4000) of FIG. 4A are enlarged in FIG. 4B to better illustrate the negative effects of the large aberrated wavefront incident thereupon.
Certain techniques for extending the dynamic range of SHWFS are available, as follows.
(1) Null Lens
Null lens partially compensates the wavefront aberration of the test optics, then can reduce wavefront deviation on the lenslet array. However, it might be necessary to fabricate a very accurate null lens for an accurate measurement. Therefore, the fabrication cost of a null lens will become prohibitively expensive. Furthermore, such null lens is designed for specific test optics, so this may not applicable for other wavefronts formed by other test optics. An example of the null lens technique is described in U.S. Pat. No. 5,233,174 to Zmek, which is incorporated herein by reference.
(2) Estimation Technique
Instead of the Null lens technique, a wavefront estimation technique is proposed for measuring strongly aberrated wavefront. One example of the estimation technique is disclosed in Ref. 1 (Michael C. Roggemann, Timothy J. Schulz, Chee W. Ngai, and Jason T. Kraft, “Joint processing of Hartmann sensor and conventional image measurements to estimate large aberrations: theory and experimental results,” Appl. Opt. 38, pp. 2249-2255 (1999)). Ref. 1 a technique for estimating aberrations that extends the strength of an aberration that may be sensed with Hartmann sensor by means of an algorithm that processes both a Hartmann sensor image and a conventional image formed with same aberration. Purportedly, strong defocus aberrations can be accurately sensed with this technique.
Another estimation technique uses Maximum Likelihood Estimation (MLE) for wavefront reconstruction as disclosed in Ref. 2 (Harrison H. Barrett, Christopher Dainty, and David Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A. 24, 391-414 (2007)). Ref. 2 describes that this technique can reduce residual wavefront error, but the MLE technique requires careful attention to all noise sources and all factors that influence the sensor data.
One of the obstacles facing the inventors of the present invention is the need to control the image quality of imaging lenses at high precision. Specifically, prior to assembling an imaging system which may be composed of a plurality of lenses, it is necessary to verify performance of each lens or group lens (unit lens). Although the aberration of the imaging system may not be so large, aberration of each group lens (unit lens) or a single lens can be large. Therefore, it is important to measure the aberration of such unit lens with the large aberration.
The Estimation technique above described requires the calculation of optical forward propagation repeatedly. Propagation model in the Ref. 1 is an angular spectrum propagation method based on the Fourier optics. The Fourier optics can calculate a propagation of the light field from surface to surface, where each surface can be a plane or a sphere. Assuming a simplified optical configuration, as shown in FIG. 5, a test lens 5010 is illuminated by a point source 5000. Light rays passing through test lens 5010 travel through a lenslet array 5020, and a wavefront thereof is detected by a CCD image sensor 5030.
For Fourier optics-based propagation calculations, a configuration shown in FIG. 6 is typically assumed. In FIG. 6, an exit pupil 6010 (representative of test optics 5010 in FIG. 5) is established at a plane P1, a lenslet array plane 6020 replaces the lenslet array 5020 at a plane P2, and a detector array plane 6030 represents the detecting surface of the CCD image sensor 5030 at plane P3. However, the simplified “plane” representation of FIG. 6 is inaccurate because the actual lenslet array 5020 is formed on a substrate that has a certain thickness, and each lenslet also has a certain thickness.
Therefore, when assuming an ideal thin plane as a lenslet array plane 6020, the accuracy of propagation calculation may be deteriorated. In particular, when light on the lenslet array 5020 is converged or diverged, the substrate of the lenslet causes large spherical aberration, and the lenslet causes coma and astigmatism aberration. Accordingly, it may be hard to implement (compensate for) these aberration effects into the Fourier optics based propagation. Furthermore, the effect of a misalignment of optical components may be also difficult to consider in the Fourier optics based method.
In addition, in the propagation model of Fourier optics, the dynamic range of the wavefront to be measured is restricted by the number of samplings required to satisfy the sampling theorem in the Fourier domain. As an example, in the configuration in FIG. 5, the distance between the point source 5000 and the test lens 5010 is 127.5 mm, and wavefront deviation on the exit pupil sphere is represented by Zernike coefficients as shown in FIG. 7.
To calculate the forward propagation model with Fourier-based optics of the test arrangement shown in FIG. 5, one can assume the Fourier plane configuration as shown in FIG. 6. Specifically, as described above, in FIG. 6, the test lens 5010 of FIG. 5 is represented by a flat exit pupil plane 6010 (dummy surface plane) at a plane P1. And wavefront deviation on the exit pupil plane 6010 (wavefront on the dummy surface plain) of FIG. 6 is shown in FIG. 8.
In this case, the number of samplings on the exit pupil plane 6010 would require around 14300 samples over the pupil plane for satisfying the sampling theorem at the outer most region of the pupil. Usually, more than twice of this number of samples is required for FFT (Fast Fourier Transform)-based propagation such as angular spectrum propagation. Thus, for the simplified configuration of FIG. 5, nearly 30000×30000 samplings might be necessary for two dimensional propagation in the Fourier optics-based method.
FIG. 9 shows estimated computational time for one FFT calculation using a processing unit (data analyzer) with a CPU AMD AM+ Phenom 9950. In case of 30000×30000 samples for FFT, this memory size is very large for a current commercial work station. If we can manipulate this size of data, it will take about 65 seconds to calculate one FFT. Since two sets of FFT calculations are necessary for the angular spectrum propagation, one angular spectrum propagation will take 130 seconds at least. In estimation procedure, a forward model is calculated iteratively. When we try to estimate several parameters simultaneously, total time for wavefront estimation becomes impractically large. When wavefront deviation becomes larger, much longer computational time and a larger memory will be required.
Therefore, the Fourier optics based propagation calculation technique requires a complicated model for the propagation calculation and intensive computational resources. As a result, this technique may only be used to estimate small number of parameters with many limitations for modeling of the propagation model in practice.